Conventionally, designing a converter begins from deriving the relationship of voltage and current. The circuit design technology is used to modify the relationship into a linear equation, wherein the carrier mobility is usually assumed to be a constant. In fact, the carrier mobility is not necessarily a constant in that the carrier mobility varies with gate voltage and that the equivalent mobility decreases when the carrier mobility reaches a saturation value. The present invention does not intend to deal with the second phenomenon because it will be solved via eliminating noise in designing an analog circuit.
Based on the voltage-current relationships in the saturation region and the linear region, the current values in the two regions are summed up with the quadratic term of the linear region being eliminated in designing a traditional voltage-current converter.
The conventional voltage-current relationships in the saturation region and the linear region may be expressed by Equation 1 and Equation 2:
                              I          lin                =                              K            lin                    ⁡                      [                                                            (                                                            V                      GS                                        -                                          V                      T                                                        )                                ⁢                                  V                  DS                                            -                                                V                  DS                  2                                2                                      ]                                              (        1        )                                          I          sat                =                              1            2                    ⁢                                                    K                sat                            ⁡                              (                                                      V                    GS                                    -                                      V                    T                                                  )                                      2                                              (        2        )            Suppose
                                          K            lin                    ⁡                      (                                          V                DS                2                            2                        )                          =                              1            2                    ⁢                                                    K                lin                            ⁡                              (                                                      V                    GS                                    -                                      V                    T                                                  )                                      2                                              (        3        )            Substitute Equation (3) into Equation (1) and Equation (2) to eliminate the quadratic term of the linear region. Then Equation (4) is obtained:Iout=K[(VGS−VT)VDS]  (4)wherein the carrier mobility K is not a constant. Refer to FIG. 1 for voltage-current relationships in the conventional technology. When K is a constant, the relationship is represented by the line segment A. However, the actual relationship is usually the line segment B. The K value of the line segment B is usually smaller than that in the line segment A because of the actions of the vertical electric field VGS, the horizontal electric field VDS, W×L, and W/L. In the analysis, the horizontal electric field VDS must be a constant. However, the K value is still affected by the vertical electric field VGS and the dimensions of the transistors. Therefore, the voltage-current relationship is not merely influenced by the quadratic term of the linear region. Whether Equation (4) really meets the requirement of the linear region is also dependent on the voltage over the element because the electric field generated by the element influences the carrier mobility (K value) and makes the output current nonlinear. The discussion of the carrier mobility (K value) will be more correct and effective if it is based on the abovementioned facts.
Because of the fact that the carrier mobility varies with the gate voltage, Equation (5) is taken into consideration:
                              μ          eff                =                              μ            0                                1            +                          θ              ⁡                              (                                                      V                    GS                                    -                                      V                    TH                                                  )                                                                        (        5        )            Substitute Equation (5) into Equation (4) to obtain Resulting Equation (1):
                    =                                            K                              1                +                                  θ                  ⁡                                      (                                                                  V                        GS                                            -                                              V                        TH                                                              )                                                                        ⁡                          [                                                (                                                            V                      GS                                        -                                          V                      T                                                        )                                ⁢                                  V                  DS                                            ]                                ≅                                    K              ⁡                              (                                                      V                    GS                                    -                                      V                    T                                                  )                                      ⁢                                          V                DS                            ⁡                              [                                                                                                    1                        -                                                  θ                          ⁡                                                      (                                                                                          V                                GS                                                            -                                                              V                                TH                                                                                      )                                                                          +                                                                                                                                                                                                                                    θ                              2                                                        ⁡                                                          (                                                                                                V                                  GS                                                                -                                                                  V                                  TH                                                                                            )                                                                                2                                                -                        …                                                                                            ]                                              ≅                                                    K                ⁡                                  (                                                            V                      GS                                        -                                          V                      T                                                        )                                            ⁢                              V                DS                                      -                          K              ⁢                                                          ⁢                                                θ                  ⁡                                      (                                                                  V                        GS                                            -                                              V                        TH                                                              )                                                  2                            ⁢                              V                DS                                      +                          K              ⁢                                                          ⁢                                                                    θ                    2                                    ⁡                                      (                                                                  V                        GS                                            -                                              V                        TH                                                              )                                                  3                            ⁢                              V                DS                                      -            …                                              Resulting        ⁢                                  ⁢        Equation        ⁢                                  ⁢                  (          1          )                    When Equation (5) including the factor of carrier mobility degradation is substituted into the linear Equation (4), it is found that Resulting Equation (1) contains not only the first order term but also the higher order terms. Thus is proved that degradation is involved in the conversion curve of voltage and current.
As carrier mobility degradation still causes the resulting equation to contain quadratic and higher-order nonlinear terms, eliminating the quadratic term of the linear region is not an optimized approach to realize a high linearity voltage-current converter. The present invention intends to decrease the higher-order nonlinear terms to an optimized amount.